The goal of this section is to present a practical scheme for the optimization of equation (REF ). The starting point is the “dualization” [1]} of the data-misfit term: \(1/2\left\Vert \mathbf {r}\right\Vert ^2_{\Sigma }=\sup _{\mathbf {y}}\langle \mathbf {y},\mathbf {r}\rangle -1/2\left\Vert \mathbf {y}\right\Vert ^2_{\Sigma ^{-1}}\) . We end up with a saddle-point problem [2]}:
\(\min _{\mathbf {m}}\max _{\mathbf {y}}\mathcal {L}(\mathbf {m},\mathbf {y})=\langle \mathbf {y},\mathbf {r}(\mathbf {m})\rangle -\dfrac{1}{2}\left\Vert \mathbf {y}\right\Vert ^2_{\widetilde{\Sigma }_{\mathrm {d}}(\mathbf {m})^{-1}},\qquad \widetilde{\Sigma }_{\mathrm {d}}(\mathbf {m})=\Sigma _{\mathrm {d}}+\mathrm {F}(\mathbf {m})\Sigma _{\mathrm {q}}\mathrm {F}(\mathbf {m})^*.\) 
